The Graph Parameter Hierarchy

The Graph Parameter Hierarchy

February 15, 2019

1 About

The purpose of this document is to gather graph parameters (also called graph invariants) and their relationships in a central place. It was edited by Manuel Sorge¹ and Mathias Weller² with contributions by Ren´e van Bevern, Florent Foucaud, Ondˇrej Such´y, Pascal Ochem, Martin Vatshelle, and Gerhard J. Woeg- inger. This project and the diagram shown inFigure 1is inspired by the work of Bart M. P. Jansen [20]. Some further related work (in no particular order) is the Information System on Graph Classes and their Inclusions (ISGCI)³, the Graph Parameter Project⁴, INGRID: a graph invariant manipulator [12] and the work by Martin Vatshelle [26].

2 Terminology

A graph parameter is a functionφ:G→RwhereGis the set of finite graphs andRis the set of real numbers. Letφ, ψgraph parameters. The parameterφ upper bounds another parameter ψ, if there is some function f such that for every graphGinGit holds thatψ(G)≤f(φ(G)); we writeφψ. Parameterφ is unbounded in ψ if ¬(ψ φ). Parameter φ strictly upper bounds ψ if φ ψ∧ ¬(ψφ). If (¬(φ ψ))∧ ¬(ψφ) then φand ψ are incomparable. If φψ∧ψφthenφandψ areequal.

1manuel.sorge@gmail.com

2mathias.weller@u-pem.fr

3http://www.graphclasses.org

4https://robert.sasak.sk/gpp/index.php

Distance to Clique Vertex Cover Max Leaf #

Minimum Clique Cover

Distance to Co-Cluster

Distance to Cluster

Distance to Disjoint Paths

4.10 Treedepth 4.28

Feedback

Edge Set Bandwidth 4.25

Maximum Independent Set

4.26

Distance to Cograph

Distance to Interval

Feedback

Vertex Set Pathwidth 4.27

Maximum Degree

4.22

Bisection Width

Minimum Dominating Set

Distance to Chordal

4.20

Distance to Bipartite

Distance to

Outerplanar Genus h-index

4.19

Max Diameter of Components

Distance to Perfect

Treewidth

Acyclic Chromatic #

4.8 4.7

4.21 Average

Distance

Degeneracy Boxicity 4.18

4.16

4.12 4.9

Cliquewidth 4.17

4.23

Chordality as4.16

Girth 4.15 4.13

Chromatic #

Average Degree

Minimum Degree

2

3 Parameter Definitions

3.1 Acyclic Chromatic Number

The acyclic chromatic number of a graphG= (V, E) is the smallest size of a vertex partitionP ={V1, . . . , V`} such that eachVi is an independent set and for allVi, Vj the graphG[Vi∪Vj] does not contain a cycle. In other words, the acyclic chromatic number is the smallest number of colors needed for a proper vertex coloring such that every two-chromatic subgraph is acyclic.

Introduced by Gr¨unbaum [18].

3.2 Covering Parameters

3.2.1 Path Number

The path number of a graphGis the minimum number of paths the edges ofG can be partitioned into [2]. Exists in “disjoint” and “overlapping” versions where the paths have to be disjoint or not, respectively.

3.2.2 Arboricity

Thearboricity of a graphGis the minimum number of forests the edges ofG can be partitioned into. It is called linear arboricity if the forests are linear (collection of paths).

3.2.3 Vertex Arboricity

The vertex arboricity (or “point arboricity”) of a graph G is the minimum number of vertex subsetsV_iofGsuch thatG[V_i] induces a forest for eachi. It is calledlinear vertex arboricity if the forests are linear (collection of paths). IfG is the line graph ofG⁰, then this equals the (linear) arboricity ofG⁰ [2].

3.2.4 Average Degree

Theaverage degree of a graphG= (V, E) is 2|E|/|V|.

3.3 Graph Intersection Parameters

LetG be a class of graphs and letG= (V, E) not necessarily inG. Letpbe the smallest numberpof setsEi withE=T

i≤pEi and each (V, Ei)∈ G. Then,pis calledG’sG-intersection number.

3.3.1 Interval-Intersection (Boxicity)

Theboxicity is theG-intersection number forG being the class of interval graphs.

Exceptionally, each clique has boxicity 0.

An equivalent alternative definition is the following. An axis-parallel b- dimensional box is a Cartesian product R1 ×R2 ×. . . ×Rb where Ri (for

1≤i≤b) is a closed interval of the form [ai, bi] on the real line. For a graphG, its boxicity is the minimum dimension b such that G is representable as the intersection graph of (axis-parallel) boxes inb-dimensional space.

3.3.2 Chordal-Intersection (Chordality)

The chordality is theG-intersection number for G being the class of chordal graphs.

3.4 Average Distance

Theaverage distanceof a graphG= (V, E) is 1/ ⁿ₂

·P

u,v∈V d(u, v), whered(u, v) is the length of a shortest path betweenuandv in G.

3.5 Bandwidth

Thebandwidth bw of a graphGis the maximum “length” of an edge in a one dimensional layout ofG. Formally:

ml := min

i:V→N{ max

{u,v}∈E{|i(u)−i(v)|}:iis injective}

3.6 Bisection Width

The width of a bipartition of a graph is the number of edges going between the parts. The bisection width of a graph G= (V, E) is the smallest width of a bipartition ofGsuch that the difference of the parts’ numbers of vertices is at most one.

3.7 Branchwidth

Abranch decomposition of a hypergraph H= (V,F) is a tuple (T, τ), whereT is a rooted binary tree and whereτ is a bijection from the leaves of T to the hyperedges F. Theorder of an edgee inT is the number of vertices v in H such that there are leavest₁, t₂ in different connected components ofT \efor whichτ(t₁), τ(t₂) both are incident tov. Thewidth of a branch decomposition is the maximum order of edges inT. The branchwidth ofH is the minimum width of branch-decompositions ofH.

3.8 Chromatic Number

Thechromatic number χof a graphGis the smallest number isuch that the the vertices ofGcan be partitioned into iindependent sets.

3.9 Cliquewidth

Letq be a positive integer. We call (G, λ) aq-labeled graphifGis a graph and λ: V(G)→ {1,2, . . . , q} is a mapping. The number λ(v) is called label of a vertex v. We introduce the following operations on labeled graphs:

(1) For everyi in{1, . . . , q}, we let•i denote the graph with only one vertex that is labeled byi(a constant operation).

(2) For every distincti andj from{1,2, . . . , q}, we define a unary operator ηi,j such that ηi,j(G, λ) = (G⁰, λ), whereV(G⁰) = V(G), and E(G⁰) = E(G)∪ {vw:v, w∈V, λ(v) =i, λ(w) =j}. In other words, the operator adds all edges between label-i vertices and label-j vertices.

(3) For every distinct i and j from {1,2, . . . , q}, we let ρ_i→j be the unary operator such thatρ_i→j(G, λ) = (G, λ⁰), whereλ⁰(v) =j ifλ(v) =i, and λ⁰(v) =λ(v) otherwise. The operator only changes the labeling so that the vertices that originally had labeli will now have labelj.

(4) Finally,⊕is a binary operation that makes the disjoint union, while keeping the labels of the vertices unchanged. Note explicitly that the union is disjoint in the sense that (G, λ)⊕(G, λ) has twice the number of vertices ofG.

Aq-expression is a well-formed expressionϕwritten with these symbols. The q-labeled graph produced by performing these operations in order therefore has a vertex for each occurrence of the constant symbol in ϕ; and thisq-labeled graph (and anyq-labeled graph isomorphic to it) is called thevalue val(ϕ) ofϕ.

If aq-expressionϕhas value (G, λ), we say that ϕis aq-expression of G. The cliquewidth of a graphG, denoted bycwd(G), is the minimum qsuch that there

is aq-expression ofG.

Cliquewidth has been defined by Courcelle and Olariu [7]. The definition above is inspired by Hlinˇen´y et al. [19].

3.10 Clique Cover

Aclique cover of a graph G= (V, E) is a partitionP ofV such that each part inP induces a clique inG. The minimum clique cover of Gis a clique cover with a minimum number of parts. Note that the clique cover number of a graph is exactly the chromatic number of its complement.

3.11 Coloring Number

Is one larger than the degeneracy. Introduced by Erd˝os and Hajnal [13].

3.12 Degeneracy

The degeneracy of a graphGis the maximum, with respect to all subgraphsG⁰ ofG, of the minimum degree of G⁰. Equivalent definitions include the minimum

outdegree over all acyclic orientations ofG and the minimum, over all linear ordering so the vertices, of the maximum, over all verticesv, of the number of neighbors ofv that occur later in the ordering.

3.13 Density

Also known as average degree.

3.14 Distance to Π

The distance toΠ of a graphG= (V, E) is the minimum size of a setX ⊆V such thatG[V \X] ∈Π, where Π is a graph class, e. g. chordal, bipartite, or clique.

3.15 Domatic Number

Thedomatic number of a graphGis the maximum number of pairwise disjoint dominating sets ofG.

3.16 h-Index

The h-index of a graph Gis the maximum integer hsuch that Gcontains h vertices of degree at leasth.

3.17 Genus

Thegenus g of a graphGis the minimum number of handles over all surfaces on whichGcan be embedded.

3.18 Girth

Thegirth of a graphGis the minimum numbergisuch thatGhas a cycle of lengthgi.

3.19 Linkage

Also known as degeneracy. Introduced by Kirousis and Thilikos [21].

3.20 Max Leaf Number

The max leaf number ml of a graph Gis the maximum number of leaves in a spanning tree ofG.

3.21 Subgraph-Cutset-Number

Thesubgraph-cutset-number is the maximum, with respect to all subgraphsG⁰ ofG, of the minimum vertex cut of G⁰. Here, a vertex cut in a graph Gis a set of vertices such that removing them fromGyields a disconnected graph;

considering graphs that contain only a single vertex also as disconnected.

3.22 Treedepth

Let thespan of a rooted tree be the graph constructed from that tree by adding an edge{u, v}, for each pair of vertices u, v where uis an ancestor ofv. The treedepthof a graphGis the smallest height of a rooted treeT such thatGis a subgraph of the span ofT.

3.23 Rankwidth

Introduced by Oum and Seymour [24]. For a definition, see [24] or [19].

3.24 Width

Consider a linear ordering of the vertices of a graphGand call itsorder the maximum over all verticesv of the neighbors ofv precedingv in the ordering.

The width of G a graph is the minimum order of all linear orderings of the vertices ofG.

Introduced by Freuder [16].

4 Proofs

4.1 Equality

Lemma 4.1([25]). Letβbe the branchwidth andωbe the treewidth. max{β,2} ≤ ω+ 1≤max{3β/2,2}.

Lemma 4.2([22]). Let dbe the degeneracy and wthe width. We have d=w.

Lemma 4.3. Letd be the degeneracy andc the subgraph-cutset-number. We haved≥c≥ bd/4c+ 1.

Proof. Let Gbe a graph with degeneracyd. For the first inequality, consider any subgraphG⁰ ofG. SinceGis d-degenerate, there is a vertexv inG⁰ with deg_G0(v)≤d. Cutting v’s neighbors yields a disconnected graph.

For the second inequality, we use a result of Mader [23, Korollar 1]. Namely, every graph G with n vertices and m edges that fulfills m > (2k−3)(n− k+ 1) andn≥2k−1 contains a k-vertex-connected subgraph. Consider any subgraphG⁰ of a graph with degeneracydsuch thatG⁰ has minimum degreed.

This subgraph has n≥d vertices and at leastdn/2 edges. Let us show that choosing any positivek≤ bd/4c+ 1 fulfills both conditions onGof the above

statement. This is clear for the second condition, as n ≥ d ≥ bd/2c+ 1 ≥ 2bd/4c+ 1≥2k−1. Since G⁰ has minimum degreed, it contains at leastdn/2 edges, and, hence we need to showdn/2>(2k−3)(n−k+ 1). Sincen−k+ 1 is positive and less than n, it suffices to show d/2>2k−3 which is implied bybd/4c ≥k−1. Hence,G⁰ contains ak-vertex-connected subgraph, and any of its supergraphs has subgraph-cutset-number at leastk.

Lemma 4.4([24]). Let rbe the rankwidth andq the cliquewidth. We haver≤ q≤2^1+r−1.

Lemma 4.5. Let abe the arboricity and dthe degeneracy. We have a≤d≤ 2a−1.

Proof. If the graphGhas degeneracyd, then there is an acyclic orientation with outdegree at mostd. Injectively mappingdedge sets to the outgoing edges of a vertex yield a partition intodforests because, if one of the edge sets contained a cycle, then there were two oriented paths from one vertex to another, which means that we would have assigned the same edge set to two outgoing edges of a vertex; absurd. Hence the arboricity is at mostd.

In the other direction, any subgraph of a graphGwith arboricityahas also arboricity at mosta. Thus, any subgraph has average degree at most 2a(n−1)/n and, hence, contains a vertex of degree at most 2a−1. Hence the degeneracy ofGis at most 2a−1.

4.2 Bounds

Lemma 4.6([15]). The acyclic chromatic numberχa is upper bounded by the maximum degree∆ (for every graph with ∆>4). We haveχa≤∆(∆−1)/2.

Lemma 4.7.The acyclic chromatic numberχais upper bounded by theh-indexh.

We haveχa≤h(h+ 1)/2.

Proof. LetGbe a graph and letH be the set of vertices of degree more thanh inG. Then the maximum degree ofG−H is at most h. Thus, byLemma 4.6, G−H can be acyclically colored with at mosth(h−1)/2 colors. Finally, assign each vertex inH a new color. Thus,Gcan be acyclically colored with at most h(h−1)/2 +h=h(h+ 1)/2 colors.

Lemma 4.8 ([3]). The acyclic chromatic numberχa is upper bounded by the genusg. We haveχa ≤4g+ 4.

Lemma 4.9 ([14]). The boxicity b is upper bounded by the acyclic chromatic numberχa (for every graph with χa>1). We haveb≤χa(χa−1).

Lemma 4.10 ([8, 11]). The max-leaf numbermlupper bounds the distance to disjoint pathsd. We haved≤ml−1.

Proof. Letnbe the number of vertices. From the proof of Corollary 2 in ref. [11]

it follows thatn= ml +γc whereγcis the smallest size of a connected dominating set. Clearly,n=d+f where f is the largest size of a vertex subset inducing a disjoint set of paths. By Theorem 4 in ref. [8] we have γc ≤f−1 and, hence, ml−1≥d.

Lemma 4.11 ([1]). The boxicityb is upper bounded by the maximum degree∆.

We haveb≤O(∆ log²∆).

Lemma 4.12 ([5]). The treewidth ω upper bounds the boxicity b. We have b≤ ω+ 2.

Lemma 4.13 ([4]). The average distance d upper bounds the girth g. We haved≥ng/(4n−4).

Lemma 4.14 ([9]). The max leaf numberm upper bounds the feedback vertex set sizef. We havef ≤m.

Lemma 4.15. The boxicityb upper-bounds the chordalityc. We havec≤b.

Proof. The bound follows from the fact that interval graphs are chordal. Strict- ness is shown, for example, by Chandran et al. [6] (even for bipartite graphs).

Lemma 4.16. The distance ito an interval graph upper bounds the boxicity b.

We haveb≤i+ 1.

Proof. LetG= (V, E) be a graph and letD⊆V be a vertex set such that|D|=i andG−D is an interval graph. We define an extension ofG−D and intersect it with an interval graph Gv for each vertex v ∈ D to obtain G as follows.

The extensionG⁰ = (V, E⁰) is obtained fromG by making each vertexu∈D universal, that is, by making every vertex in V adjacent to u. Since adding universal vertices to an interval graph yields an interval graph,G⁰ is an interval graph. For each vertexv∈D we defineGv= (V, Ev): Gv consists of a clique with vertex setV\{v}and an edge{u, v}for each vertexuincident tov. EachGv

is clearly an interval graph. It is also not hard to see thatE =E⁰∩T

v∈DEv. Hence the boxicity ofGis at mosti+ 1.

Lemma 4.17. The distancec to a cograph upper bounds the cliquewidthq. We haveq≤2^3+c−1.

Proof. LetG= (V, E) be a graph andD⊆V be a vertex set such that|D|=c andG−D is a cograph. Since cographs have cliquewidth at most 2 [7], by Lemma 4.4 we have that the rankwidth ofG−D is at most 2. Since G−D is obtained from Gby c vertex deletions and deleting a vertex decreases the rankwidth by at most 1 [19], we have that G has rankwidth at most 2 +c.

Applying againLemma 4.4, the cliquewidth ofGis at most 2^3+c−1.

Lemma 4.18. The acyclic chromatic number aupper bounds the degeneracyd.

We haved≤2 ^a₂

−1.

Proof. If the graph Ghas acyclic chromatic numbera, then there is a partition of its edges into at most ^a₂

forests (one for each pair of colors). Hence, the arboricity ofGis at most ^a₂

. The statement follows fromLemma 4.5.

Lemma 4.19. The feedback edge set number f upper bounds the genus g. We haveg≤f.

Proof. Fix a spanning forest. This graph has genus 0. Add the remaining edges one-by-one. After adding theith edge, the current graph is embeddable in a surface of genusi, because we can introduce a new handle for each edge.

Lemma 4.20. The feedback vertex setf upper bounds the distance to a chordal graphc. We havec≤f.

Proof. A tree is chordal.

4.3 Strict Bounds

Lemma 4.21([10]). The acyclic chromatic numberχa is strictly upper bounded by the treewidthω. We haveχa≤ω+ 1.

Proof. Starting in the root of a width-ω tree decomposition, assign colors to vertices in such a way that, for each bag B, the vertices in B have pairwise different colors. Since, for each cycleC, there is some bag containing at least three vertices ofC, all cycles have at least three colors. Strictness follows from n×n-grids having treewidthn+ 1 and acyclic chromatic number three.

Lemma 4.22. The maximum degree∆ is strictly upper bounded by the band- widthbw. We have∆≤2bw.

Proof. Letvbe a vertex in a graphGwith bandwidth bw. There are at most bw neighbors ofvto the “right” in a bandwidth layout ofGand equally as many to the “left”.

The bound is strict by an×n-grid example: Maximum degree is four, but treewidth isn+ 1 and treewidth is a lower bound for the bandwidth.

Lemma 4.23 ([7]). The cliquewidth of a graph is upper bounded by 2^ω+1+ 1 whereω is its treewidth. This bound is strict.

Involved proof. The bound is strict by a clique example.

Lemma 4.24.Theh-indexhstrictly upper bounds degeneracyd. We haved≤h.

Proof. Construct a “degeneracy ordering” as follows. First, enumerate all vertices of degree≤hin any order, then enumerate all remaining vertices in any order (observe that there are at mosthvertices of degree more thanh).

Strictness follows from a disjoint union of stars.

Proof. We show bw≤2ml. LetT be a BFS tree of the graph from some vertexv and letLi denote the vertices of distanceitov (horizontal “layers” ofT). We show for allithatThas at least|Li|leaves. For the last layerimax, this is trivially true. Now, assume thatT has at least |Li+1| leaves. If |Li| ≤ |Li+1|, thenT also has at least|Li|leaves. Assume |Li|>|Li+1|. However, since T is a tree, no vertex inL_i+1 is adjacent to two vertices inL_i. Thus, at least|Li−L_i+1| vertices in L_i are leaves in T. Thus, T has at least |Li| leaves, implying that max_i{|L_i|} ≤ml.

Now, by the definition ofL_i, the layout that simply puts all vertices inL_i before each vertex inL_i+1for allihas bandwidth at most max_i{|L_i|+|L_i+1|} ≤ 2 maxi{|Li|} ≤2ml.

Strictness follows from long caterpillars with degree-three backbone vertices.

Lemma 4.26. The minimum clique cover numberc strictly upper bounds the independence numberα.

Proof. LetGbe a graph. Then,c(G) =χ(G) whereχis the chromatic number and α(G) = ω(G) where ω is the clique number. Since there is a family of graphs with unboundedχandω= 2 (see Mycielski graph), there does not exist a functionf such that for all graphsGit holds thatc(G) =χ(G)≤f(ω(G)) = f(α(G)). On the other hand, it is trivial thatχω, implyingcα.

Lemma 4.27. The treedepth t strictly upper bounds the pathwidthp. We have p≤t.

Proof. LetGbe a graph andTbe a tree of heightt(G) such thatGis a subgraph of T. Construct a path-decomposition as follows. For each leaf v ∈ V(T), introduce a bag containing all ancestors ofv inT. Two bags are adjacent in the path-decomposition if the corresponding leavesu, vhave the property that there is no third leaf occuring betweenuandvin a post-order traversal ofT. It is not hard to check that this indeed results in a path-decomposition and that each edge is contained in a bag.

Strictness follows from the fact that a path withnvertices has pathwidth 1 and treedepth at leastblognc.

Lemma 4.28. The minimum vertex cover size v strictly upper bounds the treedeptht. We havet≤v+ 1.

Proof. Let Gbe a graph andU ⊆V(G) a vertex cover. Construct a treeT with vertex set V(G) by making U into a path, picking an endpointuof the path, and making each vertex inV(G)\U adjacent tou. Clearly, Gis a subgraph of the span ofT and hence the treedepth ofGis at most|U|+ 1.

Strictness can be seen as follows. Consider a rooted treeT withnleaves, each of which has distance exactly two from the root. Tree T has minimum vertex cover size nbut treedepth 2.

4.4 Unboundedness 4.5 Incomparability

Lemma 4.29. Boxicity and cliquewidth are incomparable.

Proof. Consider a square grid with side-lengthn. Its cliquewidth isn+ 1 [17].

However, such a grid has boxicity two: Arrange the grid in two-dimensional space such that each edge is either perpendicular or parallel to the coordinate axis. Tilt the grid by 45 degrees. Then, replace each vertex by a square such that it overlaps on the edges with the squares of the vertex’ neighbor and only with those squares.

For the other direction, consider a graphGwhich is a clique with a perfect matching{{v_i¹, v_i²}: 1≤i≤n/2}removed. This graph is a co-graph: we show that there is no induced pathP₄on four vertices. Consider any four vertices. The number of edges induced by these four vertices is minimized if they correspond to two edges of the removed matching. However, they still induce at least four edges and, thus, there is no inducedP4. It follows thatGhas cliquewidth two.

However, G has boxicity at least n/2: Let Gi = (V(G), Ei), 1 ≤ i ≤ b, be interval graphs such that E(G) = Tb

i=1Ei. All edges of Ghave to occur in eachEi, 1≤i≤b. However, no two of the matching edges, say{v_e¹, v_e²},{v_f¹, v_f²} may not occur inEi, since, otherwise,v_e¹, v_f¹, v²_e, v_f²is an induced cycle inGi and, hence, this graph is not an interval graph. Thus, the boxicity is at leastn/2.

Lemma 4.30. Distance to disjoint paths and treedepth are incomparable.

Proof. To see that the distance to disjoint path does not upper bound the treedepth, consider a path withnvertices, which has treedepth at least logn.

For the other direction, consider a collection ofnvertex-disjoint triangles.

Lemma 4.31. Cliquewidth and degeneracy are incomparable.

Proof. Since a clique has cliquewidth 2, cliquewidth does not upper bound degeneracy.

On the other hand, square grids with side-length n have cliquewidth ex- actlyn+ 1 [17]; but have degeneracy 4.

Lemma 4.32. Treewidth and h-index are incomparable.

Proof. The disjoint union of stars proves that the h-index can be arbitrarily large whereas the treewidth is 1.

A grid proves the other direction.

Lemma 4.33. Max leaf number and treedepth are incomparable.

Proof. Consider the set of paths and the set of stars.

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